# Basic Tensor Functionality¶

Aesara supports symbolic tensor expressions. When you type,

```>>> import aesara.tensor as at
>>> x = at.fmatrix()
```

the `x` is a `TensorVariable` instance.

The `at.fmatrix` object itself is an instance of `TensorType`. Aesara knows what type of variable `x` is because `x.type` points back to `at.fmatrix`.

This section explains the various ways in which a tensor variable can be created, the attributes and methods of `TensorVariable` and `TensorType`, and various basic symbolic math and arithmetic that Aesara supports for tensor variables.

In general, Aesara’s API tries to mirror NumPy’s, so, in most cases, it’s safe to assume that the basic NumPy array functions and methods will be available.

## Creation¶

Aesara provides a list of predefined tensor types that can be used to create a tensor variables. Variables can be named to facilitate debugging, and all of these constructors accept an optional `name` argument. For example, the following each produce a `TensorVariable` instance that stands for a 0-dimensional `ndarray` of integers with the name `'myvar'`:

```>>> x = at.scalar('myvar', dtype='int32')
>>> x = at.iscalar('myvar')
>>> x = at.tensor(dtype='int32', shape=(), name='myvar')
>>> from aesara.tensor.type import TensorType
>>> x = TensorType(dtype='int32', shape=())('myvar')
```

### Constructors with optional dtype¶

These are the simplest and often-preferred methods for creating symbolic variables in your code. By default, they produce floating-point variables (with dtype determined by `aesara.config.floatX`) so if you use these constructors it is easy to switch your code between different levels of floating-point precision.

`aesara.tensor.``scalar`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 0-dimensional `ndarray`

`aesara.tensor.``vector`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 1-dimensional `ndarray`

`aesara.tensor.``row`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 2-dimensional `ndarray` in which the number of rows is guaranteed to be 1.

`aesara.tensor.``col`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 2-dimensional `ndarray` in which the number of columns is guaranteed to be 1.

`aesara.tensor.``matrix`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 2-dimensional `ndarray`

`aesara.tensor.``tensor3`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 3-dimensional `ndarray`

`aesara.tensor.``tensor4`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 4-dimensional `ndarray`

`aesara.tensor.``tensor5`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 5-dimensional `ndarray`

`aesara.tensor.``tensor6`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 6-dimensional `ndarray`

`aesara.tensor.``tensor7`(name=None, dtype=config.floatX)[source]

Return a `Variable` for a 7-dimensional `ndarray`

### All Fully-Typed Constructors¶

The following `TensorType` instances are provided in the `aesara.tensor` module. They are all callable, and accept an optional `name` argument. So for example:

```x = at.dmatrix()        # creates one Variable with no name
x = at.dmatrix('x')     # creates one Variable with name 'x'
xyz = at.dmatrix('xyz') # creates one Variable with name 'xyz'
```
bscalar int8 0 () ()
bvector int8 1 (?,) (False,)
brow int8 2 (1,?) (True, False)
bcol int8 2 (?,1) (False, True)
bmatrix int8 2 (?,?) (False, False)
btensor3 int8 3 (?,?,?) (False, False, False)
btensor4 int8 4 (?,?,?,?) (False, False, False, False)
btensor5 int8 5 (?,?,?,?,?) (False, False, False, False, False)
btensor6 int8 6 (?,?,?,?,?,?) (False,) * 6
btensor7 int8 7 (?,?,?,?,?,?,?) (False,) * 7
wscalar int16 0 () ()
wvector int16 1 (?,) (False,)
wrow int16 2 (1,?) (True, False)
wcol int16 2 (?,1) (False, True)
wmatrix int16 2 (?,?) (False, False)
wtensor3 int16 3 (?,?,?) (False, False, False)
wtensor4 int16 4 (?,?,?,?) (False, False, False, False)
wtensor5 int16 5 (?,?,?,?,?) (False, False, False, False, False)
wtensor6 int16 6 (?,?,?,?,?,?) (False,) * 6
wtensor7 int16 7 (?,?,?,?,?,?,?) (False,) * 7
iscalar int32 0 () ()
ivector int32 1 (?,) (False,)
irow int32 2 (1,?) (True, False)
icol int32 2 (?,1) (False, True)
imatrix int32 2 (?,?) (False, False)
itensor3 int32 3 (?,?,?) (False, False, False)
itensor4 int32 4 (?,?,?,?) (False, False, False, False)
itensor5 int32 5 (?,?,?,?,?) (False, False, False, False, False)
itensor6 int32 6 (?,?,?,?,?,?) (False,) * 6
itensor7 int32 7 (?,?,?,?,?,?,?) (False,) * 7
lscalar int64 0 () ()
lvector int64 1 (?,) (False,)
lrow int64 2 (1,?) (True, False)
lcol int64 2 (?,1) (False, True)
lmatrix int64 2 (?,?) (False, False)
ltensor3 int64 3 (?,?,?) (False, False, False)
ltensor4 int64 4 (?,?,?,?) (False, False, False, False)
ltensor5 int64 5 (?,?,?,?,?) (False, False, False, False, False)
ltensor6 int64 6 (?,?,?,?,?,?) (False,) * 6
ltensor7 int64 7 (?,?,?,?,?,?,?) (False,) * 7
dscalar float64 0 () ()
dvector float64 1 (?,) (False,)
drow float64 2 (1,?) (True, False)
dcol float64 2 (?,1) (False, True)
dmatrix float64 2 (?,?) (False, False)
dtensor3 float64 3 (?,?,?) (False, False, False)
dtensor4 float64 4 (?,?,?,?) (False, False, False, False)
dtensor5 float64 5 (?,?,?,?,?) (False, False, False, False, False)
dtensor6 float64 6 (?,?,?,?,?,?) (False,) * 6
dtensor7 float64 7 (?,?,?,?,?,?,?) (False,) * 7
fscalar float32 0 () ()
fvector float32 1 (?,) (False,)
frow float32 2 (1,?) (True, False)
fcol float32 2 (?,1) (False, True)
fmatrix float32 2 (?,?) (False, False)
ftensor3 float32 3 (?,?,?) (False, False, False)
ftensor4 float32 4 (?,?,?,?) (False, False, False, False)
ftensor5 float32 5 (?,?,?,?,?) (False, False, False, False, False)
ftensor6 float32 6 (?,?,?,?,?,?) (False,) * 6
ftensor7 float32 7 (?,?,?,?,?,?,?) (False,) * 7
cscalar complex64 0 () ()
cvector complex64 1 (?,) (False,)
crow complex64 2 (1,?) (True, False)
ccol complex64 2 (?,1) (False, True)
cmatrix complex64 2 (?,?) (False, False)
ctensor3 complex64 3 (?,?,?) (False, False, False)
ctensor4 complex64 4 (?,?,?,?) (False, False, False, False)
ctensor5 complex64 5 (?,?,?,?,?) (False, False, False, False, False)
ctensor6 complex64 6 (?,?,?,?,?,?) (False,) * 6
ctensor7 complex64 7 (?,?,?,?,?,?,?) (False,) * 7
zscalar complex128 0 () ()
zvector complex128 1 (?,) (False,)
zrow complex128 2 (1,?) (True, False)
zcol complex128 2 (?,1) (False, True)
zmatrix complex128 2 (?,?) (False, False)
ztensor3 complex128 3 (?,?,?) (False, False, False)
ztensor4 complex128 4 (?,?,?,?) (False, False, False, False)
ztensor5 complex128 5 (?,?,?,?,?) (False, False, False, False, False)
ztensor6 complex128 6 (?,?,?,?,?,?) (False,) * 6
ztensor7 complex128 7 (?,?,?,?,?,?,?) (False,) * 7

### Plural Constructors¶

There are several constructors that can produce multiple variables at once. These are not frequently used in practice, but often used in tutorial examples to save space!

`iscalars, lscalars, fscalars, dscalars`

Return one or more scalar variables.

`ivectors, lvectors, fvectors, dvectors`

Return one or more vector variables.

`irows, lrows, frows, drows`

Return one or more row variables.

`icols, lcols, fcols, dcols`

Return one or more col variables.

`imatrices, lmatrices, fmatrices, dmatrices`

Return one or more matrix variables.

Each of these plural constructors accepts an integer or several strings. If an integer is provided, the method will return that many `Variables` and if strings are provided, it will create one `Variable` for each string, using the string as the `Variable`’s name. For example:

```# Creates three matrix `Variable`s with no names
x, y, z = at.dmatrices(3)
# Creates three matrix `Variables` named 'x', 'y' and 'z'
x, y, z = at.dmatrices('x', 'y', 'z')
```

### Custom tensor types¶

If you would like to construct a tensor variable with a non-standard broadcasting pattern, or a larger number of dimensions you’ll need to create your own `TensorType` instance. You create such an instance by passing the dtype and broadcasting pattern to the constructor. For example, you can create your own 8-dimensional tensor type

```>>> dtensor8 = TensorType(dtype='float64', shape=(None,)*8)
>>> x = dtensor8()
>>> z = dtensor8('z')
```

You can also redefine some of the provided types and they will interact correctly:

```>>> my_dmatrix = TensorType('float64', shape=(None,)*2)
>>> x = my_dmatrix()  # allocate a matrix variable
>>> my_dmatrix == dmatrix
True
```

See `TensorType` for more information about creating new types of tensors.

### Converting from Python Objects¶

Another way of creating a `TensorVariable` (a `TensorSharedVariable` to be precise) is by calling `aesara.shared()`

```x = aesara.shared(np.random.standard_normal((3, 4)))
```

This will return a shared variable whose `.value` is a NumPy `ndarray`. The number of dimensions and dtype of the `Variable` are inferred from the `ndarray` argument. The argument to `shared` will not be copied, and subsequent changes will be reflected in `x.value`.

For additional information, see the `shared()` documentation.

Finally, when you use a NumPy `ndarray` or a Python number together with `TensorVariable` instances in arithmetic expressions, the result is a `TensorVariable`. What happens to the `ndarray` or the number? Aesara requires that the inputs to all expressions be `Variable` instances, so Aesara automatically wraps them in a `TensorConstant`.

Note

Aesara makes a copy of any `ndarray` that is used in an expression, so subsequent changes to that `ndarray` will not have any effect on the Aesara expression in which they’re contained.

For NumPy `ndarrays` the dtype is given, but the static shape/broadcastable pattern must be inferred. The `TensorConstant` is given a type with a matching dtype, and a static shape/broadcastable pattern with a `1`/`True` for every shape dimension that is one and `None`/`False` for every dimension with an unknown shape.

For Python numbers, the static shape/broadcastable pattern is `()` but the dtype must be inferred. Python integers are stored in the smallest dtype that can hold them, so small constants like `1` are stored in a `bscalar`. Likewise, Python floats are stored in an `fscalar` if `fscalar` suffices to hold them perfectly, but a `dscalar` otherwise.

Note

When `config.floatX == float32` (see `config`), then Python floats are stored instead as single-precision floats.

For fine control of this rounding policy, see `aesara.tensor.basic.autocast_float`.

`aesara.tensor.``as_tensor_variable`(x, name=None, ndim=None)[source]

Turn an argument `x` into a `TensorVariable` or `TensorConstant`.

Many tensor `Op`s run their arguments through this function as pre-processing. It passes through `TensorVariable` instances, and tries to wrap other objects into `TensorConstant`.

When `x` is a Python number, the dtype is inferred as described above.

When `x` is a `list` or `tuple` it is passed through `np.asarray`

If the `ndim` argument is not `None`, it must be an integer and the output will be broadcasted if necessary in order to have this many dimensions.

Return type: `TensorVariable` or `TensorConstant`

## `TensorType` and `TensorVariable`¶

class `aesara.tensor.``TensorType`(Type)[source]

The `Type` class used to mark Variables that stand for `numpy.ndarray` values. `numpy.memmap`, which is a subclass of `numpy.ndarray`, is also allowed. Recalling to the tutorial, the purple box in the tutorial’s graph-structure figure is an instance of this class.

`shape`[source]
`A tuple of ``None`` and integer values representing the static shape associated with this`
``Type`.  ``None`` values represent unknown/non-fixed shape values.`

Note

Broadcastable tuples/values are an old Theano construct that are being phased-out in Aesara.

`broadcastable`[source]

A tuple of `True`/`False` values, one for each dimension. `True` in position `i` indicates that at evaluation-time, the `ndarray` will have size one in that `i`-th dimension. Such a dimension is called a broadcastable dimension (see Broadcasting).

The broadcastable pattern indicates both the number of dimensions and whether a particular dimension must have length one.

Here is a table mapping some broadcastable patterns to what they mean:

pattern interpretation
[] scalar
[True] 1D scalar (vector of length 1)
[True, True] 2D scalar (1x1 matrix)
[False] vector
[False, False] matrix
[False] * n nD tensor
[True, False] row (1xN matrix)
[False, True] column (Mx1 matrix)
[False, True, False] A Mx1xP tensor (a)
[True, False, False] A 1xNxP tensor (b)
[False, False, False] A MxNxP tensor (pattern of a + b)

For dimensions in which broadcasting is `False`, the length of this dimension can be one or more. For dimensions in which broadcasting is `True`, the length of this dimension must be one.

When two arguments to an element-wise operation (like addition or subtraction) have a different number of dimensions, the broadcastable pattern is expanded to the left, by padding with `True`. For example, a vector’s pattern, `[False]`, could be expanded to `[True, False]`, and would behave like a row (1xN matrix). In the same way, a matrix (```[False, False]```) would behave like a 1xNxP tensor (`[True, False, False]`).

If we wanted to create a `TensorType` representing a matrix that would broadcast over the middle dimension of a 3-dimensional tensor when adding them together, we would define it like this:

```>>> middle_broadcaster = TensorType('complex64', [False, True, False])
```
`ndim`[source]

The number of dimensions that a `Variable`’s value will have at evaluation-time. This must be known when we are building the expression graph.

`dtype`[source]

A string indicating the numerical type of the `ndarray` for which a `Variable` of this `Type` represents.

The `dtype` attribute of a `TensorType` instance can be any of the following strings.

dtype domain bits
`'int8'` signed integer 8
`'int16'` signed integer 16
`'int32'` signed integer 32
`'int64'` signed integer 64
`'uint8'` unsigned integer 8
`'uint16'` unsigned integer 16
`'uint32'` unsigned integer 32
`'uint64'` unsigned integer 64
`'float32'` floating point 32
`'float64'` floating point 64
`'complex64'` complex 64 (two float32)
`'complex128'` complex 128 (two float64)
`__init__`(self, dtype, broadcastable)[source]

If you wish to use a `Type` that is not already available (for example, a 5D tensor), you can build an appropriate `Type` by instantiating `TensorType`.

### `TensorVariable`¶

class `aesara.tensor.``TensorVariable`(Variable, _tensor_py_operators)[source]

A `Variable` type that represents symbolic tensors.

See `_tensor_py_operators` for most of the attributes and methods you’ll want to call.

class `aesara.tensor.``TensorConstant`(Variable, _tensor_py_operators)[source]

Python and NumPy numbers are wrapped in this type.

See `_tensor_py_operators` for most of the attributes and methods you’ll want to call.

class `aesara.tensor.``TensorSharedVariable`(Variable, _tensor_py_operators)[source]

This type is returned by `shared()` when the value to share is a NumPy ndarray.

See `_tensor_py_operators` for most of the attributes and methods you’ll want to call.

class `aesara.tensor.var.``_tensor_py_operators`[source]

This mix-in class adds convenient attributes, methods, and support to `TensorVariable`, `TensorConstant` and `TensorSharedVariable` for Python operators (see Operator Support).

`type`[source]

A reference to the `TensorType` instance describing the sort of values that might be associated with this variable.

`ndim`[source]
The number of dimensions of this tensor. Aliased to `TensorType.ndim`.
`dtype`[source]
The numeric type of this tensor. Aliased to `TensorType.dtype`.
`reshape`(shape, ndim=None)[source]

Returns a view of this tensor that has been reshaped as in `numpy.reshape`. If the shape is a `Variable` argument, then you might need to use the optional `ndim` parameter to declare how many elements the shape has, and therefore how many dimensions the reshaped Variable will have.

`dimshuffle`(*pattern)[source]

Returns a view of this tensor with permuted dimensions. Typically the pattern will include the integers `0, 1, ... ndim-1`, and any number of `'x'` characters in dimensions where this tensor should be broadcasted.

A few examples of patterns and their effect:

• `('x',)`: make a 0d (scalar) into a 1d vector
• `(0, 1)`: identity for 2d vectors
• `(1, 0)`: inverts the first and second dimensions
• `('x', 0)`: make a row out of a 1d vector (N to 1xN)
• `(0, 'x')`: make a column out of a 1d vector (N to Nx1)
• `(2, 0, 1)`: AxBxC to CxAxB
• `(0, 'x', 1)`: AxB to Ax1xB
• `(1, 'x', 0)`: AxB to Bx1xA
• `(1,)`: This removes the dimension at index 0. It must be a broadcastable dimension.
`flatten`(ndim=1)[source]

Returns a view of this tensor with `ndim` dimensions, whose shape for the first `ndim-1` dimensions will be the same as `self`, and shape in the remaining dimension will be expanded to fit in all the data from `self`.

`ravel`()[source]

return `flatten`. For NumPy compatibility.

`T`[source]

Transpose of this tensor.

```>>> x = at.zmatrix()
>>> y = 3+.2j * x.T
```
`{any,all}(axis=None, keepdims=False)`
`{sum,prod,mean}(axis=None, dtype=None, keepdims=False, acc_dtype=None)`
`{var,std,min,max,argmin,argmax}(axis=None, keepdims=False),`
`diagonal`(offset=0, axis1=0, axis2=1)[source]
`astype`(dtype)[source]
`take`(indices, axis=None, mode='raise')[source]
`copy`()[source]

Return a new symbolic variable that is a copy of the variable. Does not copy the tag.

`norm`(L, axis=None)[source]
`nonzero`(self, return_matrix=False)[source]
`nonzero_values`(self)[source]
`sort`(self, axis=- 1, kind='quicksort', order=None)[source]
`argsort`(self, axis=- 1, kind='quicksort', order=None)[source]
`clip(self, a_min, a_max) with a_min <= a_max`
`conf`()[source]
`repeat`(repeats, axis=None)[source]
`round`(mode='half_away_from_zero')[source]
`trace`()[source]
`get_scalar_constant_value`()[source]
`zeros_like`(model, dtype=None)[source]

All the above methods are equivalent to NumPy for Aesara on the current tensor.

`__{abs,neg,lt,le,gt,ge,invert,and,or,add,sub,mul,div,truediv,floordiv}__`

Those elemwise operation are supported via Python syntax.

`argmax`(axis=None, keepdims=False)[source]

See `aesara.tensor.math.argmax`.

`argmin`(axis=None, keepdims=False)[source]

See `aesara.tensor.math.argmin`.

`argsort`(axis=- 1, kind='quicksort', order=None)[source]

See `aesara.tensor.sort.argsort`.

property `broadcastable`[source]

The broadcastable signature of this tensor.

`broadcasting`

`choose`(choices, mode='raise')[source]

Construct an array from an index array and a set of arrays to choose from.

`clip`(a_min, a_max)[source]

See `aesara.tensor.math.clip`.

`compress`(a, axis=None)[source]

Return selected slices only.

`conj`()[source]

See `aesara.tensor.math.conj`.

`conjugate`()[source]

See `aesara.tensor.math.conj`.

`copy`(name=None)[source]

Return a symbolic copy and optionally assign a name.

Does not copy the tags.

`dimshuffle`(*pattern)[source]

Reorder the dimensions of this variable, optionally inserting broadcasted dimensions.

Parameters: pattern – List/tuple of int mixed with ‘x’ for broadcastable dimensions.

Examples

For example, to create a 3D view of a [2D] matrix, call `dimshuffle([0,'x',1])`. This will create a 3D view such that the middle dimension is an implicit broadcasted dimension. To do the same thing on the transpose of that matrix, call `dimshuffle([1, 'x', 0])`.

Notes

This function supports the pattern passed as a tuple, or as a variable-length argument (e.g. `a.dimshuffle(pattern)` is equivalent to `a.dimshuffle(*pattern)` where `pattern` is a list/tuple of ints mixed with ‘x’ characters).

`DimShuffle`

property `dtype`[source]

The dtype of this tensor.

`fill`(value)[source]

Fill inputted tensor with the assigned value.

property `imag`: Union[aesara.graph.basic.Variable, List[aesara.graph.basic.Variable]][source]

Return imaginary component of complex-valued tensor `z` Generalizes a scalar `Op` to tensors.

All the inputs must have the same number of dimensions. When the `Op` is performed, for each dimension, each input’s size for that dimension must be the same. As a special case, it can also be one but only if the input’s `broadcastable` flag is `True` for that dimension. In that case, the tensor is (virtually) replicated along that dimension to match the size of the others.

The dtypes of the outputs mirror those of the scalar `Op` that is being generalized to tensors. In particular, if the calculations for an output are done in-place on an input, the output type must be the same as the corresponding input type (see the doc of `ScalarOp` to get help about controlling the output type)

-`Elemwise(add)`: represents `+` on tensors `x + y` -`Elemwise(add, {0 : 0})`: represents the `+=` operation `x += y` -`Elemwise(add, {0 : 1})`: represents `+=` on the second argument `y += x` -`Elemwise(mul)(np.random.random((10, 5)), np.random.random((1, 5)))`: the second input is completed along the first dimension to match the first input -`Elemwise(true_div)(np.random.random(10, 5), np.random.random(10, 1))`: same but along the second dimension -`Elemwise(int_div)(np.random.random((1, 5)), np.random.random((10, 1)))`: the output has size `(10, 5)`. -`Elemwise(log)(np.random.random((3, 4, 5)))`

`max`(axis=None, keepdims=False)[source]

See `aesara.tensor.math.max`.

`mean`(axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]

See `aesara.tensor.math.mean`.

`min`(axis=None, keepdims=False)[source]

See `aesara.tensor.math.min`.

property `ndim`[source]

The rank of this tensor.

`nonzero`(return_matrix=False)[source]

See `aesara.tensor.basic.nonzero`.

`nonzero_values`()[source]

See `aesara.tensor.basic.nonzero_values`.

`prod`(axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]

See `aesara.tensor.math.prod`.

`ptp`(axis=None)[source]

See `aesara.tensor.math.ptp`.

property `real`: Union[aesara.graph.basic.Variable, List[aesara.graph.basic.Variable]][source]

Return real component of complex-valued tensor `z` Generalizes a scalar `Op` to tensors.

All the inputs must have the same number of dimensions. When the `Op` is performed, for each dimension, each input’s size for that dimension must be the same. As a special case, it can also be one but only if the input’s `broadcastable` flag is `True` for that dimension. In that case, the tensor is (virtually) replicated along that dimension to match the size of the others.

The dtypes of the outputs mirror those of the scalar `Op` that is being generalized to tensors. In particular, if the calculations for an output are done in-place on an input, the output type must be the same as the corresponding input type (see the doc of `ScalarOp` to get help about controlling the output type)

-`Elemwise(add)`: represents `+` on tensors `x + y` -`Elemwise(add, {0 : 0})`: represents the `+=` operation `x += y` -`Elemwise(add, {0 : 1})`: represents `+=` on the second argument `y += x` -`Elemwise(mul)(np.random.random((10, 5)), np.random.random((1, 5)))`: the second input is completed along the first dimension to match the first input -`Elemwise(true_div)(np.random.random(10, 5), np.random.random(10, 1))`: same but along the second dimension -`Elemwise(int_div)(np.random.random((1, 5)), np.random.random((10, 1)))`: the output has size `(10, 5)`. -`Elemwise(log)(np.random.random((3, 4, 5)))`

`repeat`(repeats, axis=None)[source]

See `aesara.tensor.basic.repeat`.

`reshape`(shape, ndim=None)[source]

Return a reshaped view/copy of this variable.

Parameters: shape – Something that can be converted to a symbolic vector of integers. ndim – The length of the shape. Passing None here means for Aesara to try and guess the length of `shape`.

Warning

This has a different signature than numpy’s ndarray.reshape! In numpy you do not need to wrap the shape arguments in a tuple, in aesara you do need to.

`round`(mode=None)[source]

See `aesara.tensor.math.round`.

`sort`(axis=- 1, kind='quicksort', order=None)[source]

See `aesara.tensor.sort.sort`.

`squeeze`()[source]

Remove broadcastable dimensions from the shape of an array.

It returns the input array, but with the broadcastable dimensions removed. This is always `x` itself or a view into `x`.

`std`(axis=None, ddof=0, keepdims=False, corrected=False)[source]

See `aesara.tensor.math.std`.

`sum`(axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]

See `aesara.tensor.math.sum`.

`swapaxes`(axis1, axis2)[source]

See `aesara.tensor.basic.swapaxes`.

If a matrix is provided with the right axes, its transpose will be returned.

`transfer`(target)[source]

Transfer this this array’s data to another device.

If `target` is `'cpu'` this will transfer to a TensorType (if not already one). Other types may define additional targets.

Parameters: target (str) – The desired location of the output variable
`transpose`(*axes)[source]

Transpose this array.

Returns: object – `tensor.transpose(self, axes)` or `tensor.transpose(self, axes)`. If only one `axes` argument is provided and it is iterable, then it is assumed to be the entire axes tuple, and passed intact to tensor.transpose.
`var`(axis=None, ddof=0, keepdims=False, corrected=False)[source]

See `aesara.tensor.math.var`.

## Shaping and Shuffling¶

To re-order the dimensions of a variable, to insert or remove broadcastable dimensions, see `_tensor_py_operators.dimshuffle()`.

`aesara.tensor.``shape`(x)[source]

Returns an `lvector` representing the shape of `x`.

`aesara.tensor.``reshape`(x, newshape, ndim=None)[source]
type x: any `TensorVariable` (or compatible) variable to be reshaped `lvector` (or compatible) the new shape for `x` optional - the length that `newshape`’s value will have. If this is `None`, then `reshape` will infer it from `newshape`. variable with `x`’s dtype, but `ndim` dimensions

Note

This function can infer the length of a symbolic `newshape` value in some cases, but if it cannot and you do not provide the `ndim`, then this function will raise an Exception.

`aesara.tensor.``shape_padleft`(x, n_ones=1)[source]

Reshape `x` by left padding the shape with `n_ones` 1s. Note that all this new dimension will be broadcastable. To make them non-broadcastable see the `unbroadcast()`.

Parameters: x (any `TensorVariable` (or compatible)) – variable to be reshaped
`aesara.tensor.``shape_padright`(x, n_ones=1)[source]

Reshape `x` by right padding the shape with `n_ones` ones. Note that all this new dimension will be broadcastable. To make them non-broadcastable see the `unbroadcast()`.

Parameters: x (any TensorVariable (or compatible)) – variable to be reshaped
`aesara.tensor.``shape_padaxis`(t, axis)[source]

Reshape `t` by inserting `1` at the dimension `axis`. Note that this new dimension will be broadcastable. To make it non-broadcastable see the `unbroadcast()`.

Parameters: x (any `TensorVariable` (or compatible)) – variable to be reshaped axis (int) – axis where to add the new dimension to `x`

Example:

```>>> tensor = aesara.tensor.type.tensor3()
InplaceDimShuffle{x,0,1,2}.0
InplaceDimShuffle{0,x,1,2}.0
InplaceDimShuffle{0,1,2,x}.0
InplaceDimShuffle{0,1,2,x}.0
```
`aesara.tensor.``unbroadcast`(x, *axes)[source]

Make the input impossible to broadcast in the specified axes.

For example, addbroadcast(x, 0) will make the first dimension of x broadcastable. When performing the function, if the length of x along that dimension is not 1, a ValueError will be raised.

We apply the opt here not to pollute the graph

Parameters: x (tensor_like) – Input aesara tensor. axis (an int or an iterable object such as list or tuple of int values) – The dimension along which the tensor x should be unbroadcastable. If the length of x along these dimensions is not 1, a ValueError will be raised. A aesara tensor, which is unbroadcastable along the specified dimensions. tensor
`aesara.tensor.``addbroadcast`(x, *axes)[source]

Make the input broadcastable in the specified axes.

For example, addbroadcast(x, 0) will make the first dimension of x broadcastable. When performing the function, if the length of x along that dimension is not 1, a ValueError will be raised.

We apply the opt here not to pollute the graph

Parameters: x (tensor_like) – Input aesara tensor. axis (an int or an iterable object such as list or tuple of int values) – The dimension along which the tensor x should be broadcastable. If the length of x along these dimensions is not 1, a ValueError will be raised. A aesara tensor, which is broadcastable along the specified dimensions. tensor
`aesara.tensor.``patternbroadcast`(x, broadcastable)[source]

For example, `patternbroadcast(x, (True, False))` will make the first dimension of `x` broadcastable and the second dimension not broadcastable, so `x` will now be a row.

Parameters: x – Input to re-broadcast. broadcastable – Truthy values indicating whether or not a dimension should be broadcastable or not. If the length of `x` along these dimensions is not `1`, a `ValueError` will be raised.
`aesara.tensor.``flatten`(x, ndim=1)[source]

Similar to `reshape()`, but the shape is inferred from the shape of `x`.

Parameters: x (any `TensorVariable` (or compatible)) – variable to be flattened ndim (int) – the number of dimensions in the returned variable variable with same dtype as `x` and `ndim` dimensions variable with the same shape as `x` in the leading `ndim-1` dimensions, but with all remaining dimensions of `x` collapsed into the last dimension.

For example, if we flatten a tensor of shape `(2, 3, 4, 5)` with ```flatten(x, ndim=2)```, then we’ll have the same (i.e. `2-1=1`) leading dimensions `(2,)`, and the remaining dimensions are collapsed, so the output in this example would have shape `(2, 60)`.

`aesara.tensor.``tile`(x, reps, ndim=None)[source]

Construct an array by repeating the input `x` according to `reps` pattern.

Tiles its input according to `reps`. The length of `reps` is the number of dimension of `x` and contains the number of times to tile `x` in each dimension.

See: numpy.tile documentation for examples. `aesara.tensor.extra_ops.repeat` Currently, `reps` must be a constant, `x.ndim` and `len(reps)` must be equal and, if specified, `ndim` must be equal to both.
`aesara.tensor.``roll`(x, shift, axis=None)[source]

Convenience function to roll TensorTypes along the given axis.

Syntax copies numpy.roll function.

Parameters: x (tensor_like) – Input tensor. shift (int (symbolic or literal)) – The number of places by which elements are shifted. axis (int (symbolic or literal), optional) – The axis along which elements are shifted. By default, the array is flattened before shifting, after which the original shape is restored. Output tensor, with the same shape as `x`. tensor

## Creating Tensors¶

`aesara.tensor.``zeros_like`(x, dtype=None)[source]
Parameters: x – tensor that has the same shape as output dtype – data-type, optional By default, it will be x.dtype.

Returns a tensor the shape of `x` filled with zeros of the type of `dtype`.

`aesara.tensor.``ones_like`(x)[source]
Parameters: x – tensor that has the same shape as output dtype – data-type, optional By default, it will be `x.dtype`.

Returns a tensor the shape of `x` filled with ones of the type of `dtype`.

`aesara.tensor.``zeros`(shape, dtype=None)[source]
Parameters: shape – a tuple/list of scalar with the shape information. dtype – the dtype of the new tensor. If `None`, will use `"floatX"`.

Returns a tensor filled with zeros of the provided shape.

`aesara.tensor.``ones`(shape, dtype=None)[source]
Parameters: shape – a tuple/list of scalar with the shape information. dtype – the dtype of the new tensor. If `None`, will use `"floatX"`.

Returns a tensor filled with ones of the provided shape.

`aesara.tensor.``fill`(a, b)[source]
Parameters: a – tensor that has same shape as output b – Aesara scalar or value with which you want to fill the output

Create a matrix by filling the shape of `a` with `b`.

`aesara.tensor.``alloc`(value, *shape)[source]
Parameters: value – a value with which to fill the output shape – the dimensions of the returned array an N-dimensional tensor initialized by `value` and having the specified shape.
`aesara.tensor.``eye`(n, m=None, k=0, dtype=aesara.config.floatX)[source]
Parameters: n – number of rows in output (value or Aesara scalar) m – number of columns in output (value or Aesara scalar) k – Index of the diagonal: `0` refers to the main diagonal, a positive value refers to an upper diagonal, and a negative value to a lower diagonal. It can be an Aesara scalar. An array where all elements are equal to zero, except for the `k`-th diagonal, whose values are equal to one.
`aesara.tensor.``identity_like`(x)[source]
Parameters: x – tensor A tensor of same shape as `x` that is filled with zeros everywhere except for the main diagonal, whose values are equal to one. The output will have same dtype as `x`.
`aesara.tensor.``stack`(tensors, axis=0)[source]

Stack tensors in sequence on given axis (default is `0`).

Take a sequence of tensors and stack them on given axis to make a single tensor. The size in dimension `axis` of the result will be equal to the number of tensors passed.

Parameters: tensors – a list or a tuple of one or more tensors of the same rank. axis – the axis along which the tensors will be stacked. Default value is `0`. A tensor such that `rval == tensors`, `rval == tensors`, etc.

Examples:

```>>> a = aesara.tensor.type.scalar()
>>> b = aesara.tensor.type.scalar()
>>> c = aesara.tensor.type.scalar()
>>> x = aesara.tensor.stack([a, b, c])
>>> x.ndim # x is a vector of length 3.
1
>>> a = aesara.tensor.type.tensor4()
>>> b = aesara.tensor.type.tensor4()
>>> c = aesara.tensor.type.tensor4()
>>> x = aesara.tensor.stack([a, b, c])
>>> x.ndim # x is a 5d tensor.
5
>>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c]))
>>> rval.shape # 3 tensors are stacked on axis 0
(3, 2, 2, 2, 2)
```

We can also specify different axis than default value `0`:

```>>> x = aesara.tensor.stack([a, b, c], axis=3)
>>> x.ndim
5
>>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c]))
>>> rval.shape # 3 tensors are stacked on axis 3
(2, 2, 2, 3, 2)
>>> x = aesara.tensor.stack([a, b, c], axis=-2)
>>> x.ndim
5
>>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c]))
>>> rval.shape # 3 tensors are stacked on axis -2
(2, 2, 2, 3, 2)
```
`aesara.tensor.``stack`(*tensors)[source]

Warning

The interface `stack(*tensors)` is deprecated! Use `stack(tensors, axis=0)` instead.

Stack tensors in sequence vertically (row wise).

Take a sequence of tensors and stack them vertically to make a single tensor.

param tensors: one or more tensors of the same rank A tensor such that `rval == tensors`, `rval == tensors`, etc.
```>>> x0 = at.scalar()
>>> x1 = at.scalar()
>>> x2 = at.scalar()
>>> x = at.stack(x0, x1, x2)
>>> x.ndim # x is a vector of length 3.
1
```
`aesara.tensor.``concatenate`(tensor_list, axis=0)[source]
Parameters: tensor_list (a list or tuple of Tensors that all have the same shape in the axes not specified by the `axis` argument.) – one or more Tensors to be concatenated together into one. axis (literal or symbolic integer) – Tensors will be joined along this axis, so they may have different `shape[axis]`
```>>> x0 = at.fmatrix()
>>> x1 = at.ftensor3()
>>> x2 = at.fvector()
>>> x = at.concatenate([x0, x1, at.shape_padright(x2)], axis=1)
>>> x.ndim
2
```
`aesara.tensor.``stacklists`(tensor_list)[source]
Parameters: tensor_list (an iterable that contains either tensors or other iterables of the same type as `tensor_list` (in other words, this is a tree whose leaves are tensors).) – tensors to be stacked together.

Recursively stack lists of tensors to maintain similar structure.

This function can create a tensor from a shaped list of scalars:

```>>> from aesara.tensor import stacklists, scalars, matrices
>>> from aesara import function
>>> a, b, c, d = scalars('abcd')
>>> X = stacklists([[a, b], [c, d]])
>>> f = function([a, b, c, d], X)
>>> f(1, 2, 3, 4)
array([[ 1.,  2.],
[ 3.,  4.]])
```

We can also stack arbitrarily shaped tensors. Here we stack matrices into a 2 by 2 grid:

```>>> from numpy import ones
>>> a, b, c, d = matrices('abcd')
>>> X = stacklists([[a, b], [c, d]])
>>> f = function([a, b, c, d], X)
>>> x = ones((4, 4), 'float32')
>>> f(x, x, x, x).shape
(2, 2, 4, 4)
```
`aesara.tensor.basic.``choose`(a, choices, mode='raise')[source]

Construct an array from an index array and a set of arrays to choose from.

First of all, if confused or uncertain, definitely look at the Examples - in its full generality, this function is less simple than it might seem from the following code description (below ndi = numpy.lib.index_tricks):

np.choose(a,c) == np.array([c[a[I]][I] for I in ndi.ndindex(a.shape)]).

But this omits some subtleties. Here is a fully general summary:

Given an `index` array (a) of integers and a sequence of n arrays (choices), a and each choice array are first broadcast, as necessary, to arrays of a common shape; calling these Ba and Bchoices[i], i = 0,…,n-1 we have that, necessarily, Ba.shape == Bchoices[i].shape for each i. Then, a new array with shape Ba.shape is created as follows:

• if mode=raise (the default), then, first of all, each element of a (and thus Ba) must be in the range [0, n-1]; now, suppose that i (in that range) is the value at the (j0, j1, …, jm) position in Ba - then the value at the same position in the new array is the value in Bchoices[i] at that same position;
• if mode=wrap, values in a (and thus Ba) may be any (signed) integer; modular arithmetic is used to map integers outside the range [0, n-1] back into that range; and then the new array is constructed as above;
• if mode=clip, values in a (and thus Ba) may be any (signed) integer; negative integers are mapped to 0; values greater than n-1 are mapped to n-1; and then the new array is constructed as above.
Parameters: a (int array) – This array must contain integers in [0, n-1], where n is the number of choices, unless mode=wrap or mode=clip, in which cases any integers are permissible. choices (sequence of arrays) – Choice arrays. a and all of the choices must be broadcastable to the same shape. If choices is itself an array (not recommended), then its outermost dimension (i.e., the one corresponding to choices.shape) is taken as defining the `sequence`. mode ({`raise` (default), `wrap`, `clip`}, optional) – Specifies how indices outside [0, n-1] will be treated: `raise` : an exception is raised `wrap` : value becomes value mod n `clip` : values < 0 are mapped to 0, values > n-1 are mapped to n-1 The merged result. merged_array - array ValueError - shape mismatch – If a and each choice array are not all broadcastable to the same shape.

## Reductions¶

`aesara.tensor.``max`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the maximum keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. maximum of x along axis
axis can be:
• None - in which case the maximum is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``argmax`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis along which to compute the index of the maximum keepdims - (boolean) If this is set to True, the axis which is reduced is left in the result as a dimension with size one. With this option, the result will broadcast correctly against the original tensor. the index of the maximum value along a given axis

if `axis == None`, `argmax` over the flattened tensor (like NumPy)

`aesara.tensor.``max_and_argmax`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis along which to compute the maximum and its index keepdims - (boolean) If this is set to True, the axis which is reduced is left in the result as a dimension with size one. With this option, the result will broadcast correctly against the original tensor. the maximum value along a given axis and its index.

if `axis == None`, `max_and_argmax` over the flattened tensor (like NumPy)

`aesara.tensor.``min`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the minimum keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. minimum of x along axis
`axis` can be:
• `None` - in which case the minimum is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``argmin`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis along which to compute the index of the minimum keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. the index of the minimum value along a given axis

if `axis == None`, `argmin` over the flattened tensor (like NumPy)

`aesara.tensor.``sum`(x, axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the sum dtype - The dtype of the returned tensor. If None, then we use the default dtype which is the same as the input tensor’s dtype except when: the input dtype is a signed integer of precision < 64 bit, in which case we use int64 the input dtype is an unsigned integer of precision < 64 bit, in which case we use uint64 This default dtype does _not_ depend on the value of “acc_dtype”. keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. acc_dtype - The dtype of the internal accumulator. If None (default), we use the dtype in the list below, or the input dtype if its precision is higher: for int dtypes, we use at least int64; for uint dtypes, we use at least uint64; for float dtypes, we use at least float64; for complex dtypes, we use at least complex128. sum of x along axis
`axis` can be:
• `None` - in which case the sum is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``prod`(x, axis=None, dtype=None, keepdims=False, acc_dtype=None, no_zeros_in_input=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the product dtype - The dtype of the returned tensor. If None, then we use the default dtype which is the same as the input tensor’s dtype except when: the input dtype is a signed integer of precision < 64 bit, in which case we use int64 the input dtype is an unsigned integer of precision < 64 bit, in which case we use uint64 This default dtype does _not_ depend on the value of “acc_dtype”. keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. acc_dtype - The dtype of the internal accumulator. If None (default), we use the dtype in the list below, or the input dtype if its precision is higher: for int dtypes, we use at least int64; for uint dtypes, we use at least uint64; for float dtypes, we use at least float64; for complex dtypes, we use at least complex128. no_zeros_in_input - The grad of prod is complicated as we need to handle 3 different cases: without zeros in the input reduced group, with 1 zero or with more zeros. This could slow you down, but more importantly, we currently don’t support the second derivative of the 3 cases. So you cannot take the second derivative of the default prod(). To remove the handling of the special cases of 0 and so get some small speed up and allow second derivative set `no_zeros_in_inputs` to `True`. It defaults to `False`. It is the user responsibility to make sure there are no zeros in the inputs. If there are, the grad will be wrong. product of every term in x along axis
`axis` can be:
• `None` - in which case the sum is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``mean`(x, axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the mean dtype - The dtype to cast the result of the inner summation into. For instance, by default, a sum of a float32 tensor will be done in float64 (acc_dtype would be float64 by default), but that result will be casted back in float32. keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. acc_dtype - The dtype of the internal accumulator of the inner summation. This will not necessarily be the dtype of the output (in particular if it is a discrete (int/uint) dtype, the output will be in a float type). If None, then we use the same rules as `sum()`. mean value of x along axis
`axis` can be:
• `None` - in which case the mean is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``var`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the variance keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. variance of x along axis
`axis` can be:
• `None` - in which case the variance is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``std`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to compute the standard deviation keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. variance of x along axis
`axis` can be:
• `None` - in which case the standard deviation is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``all`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to apply ‘bitwise and’ keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. bitwise and of x along axis
`axis` can be:
• `None` - in which case the ‘bitwise and’ is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``any`(x, axis=None, keepdims=False)[source]
Parameter: x - symbolic Tensor (or compatible) axis - axis or axes along which to apply bitwise or keepdims - (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. bitwise or of x along axis
`axis` can be:
• `None` - in which case the ‘bitwise or’ is computed along all axes (like NumPy)
• an int - computed along this axis
• a list of ints - computed along these axes
`aesara.tensor.``ptp`(x, axis=None)[source]

Range of values (maximum - minimum) along an axis. The name of the function comes from the acronym for peak to peak.

Parameter: x Input tensor. axis Axis along which to find the peaks. By default, flatten the array. A new array holding the result.

## Indexing¶

Like NumPy, Aesara distinguishes between basic and advanced indexing. Aesara fully supports basic indexing (see NumPy’s indexing) and integer advanced indexing.

Index-assignment is not supported. If you want to do something like ```a = b``` or `a+=b`, see `aesara.tensor.subtensor.set_subtensor()` and `aesara.tensor.subtensor.inc_subtensor()` below.

`aesara.tensor.subtensor.``set_subtensor`(x, y, inplace=False, tolerate_inplace_aliasing=False)[source]

Return x with the given subtensor overwritten by y.

Parameters: x – Symbolic variable for the lvalue of = operation. y – Symbolic variable for the rvalue of = operation. tolerate_inplace_aliasing – See inc_subtensor for documentation.

Examples

To replicate the numpy expression “r[10:] = 5”, type

```>>> r = ivector()
>>> new_r = set_subtensor(r[10:], 5)
```
`aesara.tensor.subtensor.``inc_subtensor`(x, y, inplace=False, set_instead_of_inc=False, tolerate_inplace_aliasing=False, ignore_duplicates=False)[source]

Update the value of an indexed array by a given amount.

This is equivalent to `x[indices] += y` or `np.add.at(x, indices, y)`, depending on the value of `ignore_duplicates`.

Parameters: x – The symbolic result of a Subtensor operation. y – The amount by which to increment the array. inplace – Don’t use. Aesara will do in-place operations itself, when possible. set_instead_of_inc – If True, do a set_subtensor instead. tolerate_inplace_aliasing – Allow `x` and `y` to be views of a single underlying array even while working in-place. For correct results, `x` and `y` must not be overlapping views; if they overlap, the result of this `Op` will generally be incorrect. This value has no effect if `inplace=False`. ignore_duplicates – This determines whether or not `x[indices] += y` is used or `np.add.at(x, indices, y)`. When the special duplicates handling of `np.add.at` isn’t required, setting this option to `True` (i.e. using `x[indices] += y`) can resulting in faster compiled graphs.

Examples

To replicate the expression `r[10:] += 5`:

..code-block:: python

r = ivector() new_r = inc_subtensor(r[10:], 5)

To replicate the expression `r[[0, 1, 0]] += 5`:

..code-block:: python

r = ivector() new_r = inc_subtensor(r[10:], 5, ignore_duplicates=True)

## Operator Support¶

Many Python operators are supported.

```>>> a, b = at.itensor3(), at.itensor3() # example inputs
```

### Arithmetic¶

```>>> a + 3      # at.add(a, 3) -> itensor3
>>> 3 - a      # at.sub(3, a)
>>> a * 3.5    # at.mul(a, 3.5) -> ftensor3 or dtensor3 (depending on casting)
>>> 2.2 / a    # at.truediv(2.2, a)
>>> 2.2 // a   # at.intdiv(2.2, a)
>>> 2.2**a     # at.pow(2.2, a)
>>> b % a      # at.mod(b, a)
```

### Bitwise¶

```>>> a & b      # at.and_(a,b)    bitwise and (alias at.bitwise_and)
>>> a ^ 1      # at.xor(a,1)     bitwise xor (alias at.bitwise_xor)
>>> a | b      # at.or_(a,b)     bitwise or (alias at.bitwise_or)
>>> ~a         # at.invert(a)    bitwise invert (alias at.bitwise_not)
```

### Inplace¶

In-place operators are not supported. Aesara’s graph-optimizations will determine which intermediate values to use for in-place computations. If you would like to update the value of a shared variable, consider using the `updates` argument to `Aesara.function()`.

## `Elemwise`¶

### Casting¶

`aesara.tensor.``cast`(x, dtype)[source]

Cast any tensor `x` to a tensor of the same shape, but with a different numerical type `dtype`.

This is not a reinterpret cast, but a coercion `cast`, similar to `numpy.asarray(x, dtype=dtype)`.

```import aesara.tensor as at
x = at.matrix()
x_as_int = at.cast(x, 'int32')
```

Attempting to casting a complex value to a real value is ambiguous and will raise an exception. Use `real`, `imag`, `abs`, or `angle`.

`aesara.tensor.``real`(x)[source]

Return the real (not imaginary) components of tensor `x`. For non-complex `x` this function returns `x`.

`aesara.tensor.``imag`(x)[source]

Return the imaginary components of tensor `x`. For non-complex `x` this function returns `zeros_like(x)`.

### Comparisons¶

The six usual equality and inequality operators share the same interface.
Parameter: a - symbolic Tensor (or compatible) b - symbolic Tensor (or compatible) symbolic Tensor a symbolic tensor representing the application of the logical `Elemwise` operator.

Note

Aesara has no boolean dtype. Instead, all boolean tensors are represented in `'int8'`.

Here is an example with the less-than operator.

```import aesara.tensor as at
x,y = at.dmatrices('x','y')
z = at.le(x,y)
```
`aesara.tensor.``lt`(a, b)[source]

Returns a symbolic `'int8'` tensor representing the result of logical less-than (a<b).

Also available using syntax `a < b`

`aesara.tensor.``gt`(a, b)[source]

Returns a symbolic `'int8'` tensor representing the result of logical greater-than (a>b).

Also available using syntax `a > b`

`aesara.tensor.``le`(a, b)[source]

Returns a variable representing the result of logical less than or equal (a<=b).

Also available using syntax `a <= b`

`aesara.tensor.``ge`(a, b)[source]

Returns a variable representing the result of logical greater or equal than (a>=b).

Also available using syntax `a >= b`

`aesara.tensor.``eq`(a, b)[source]

Returns a variable representing the result of logical equality (a==b).

`aesara.tensor.``neq`(a, b)[source]

Returns a variable representing the result of logical inequality (a!=b).

`aesara.tensor.``isnan`(a)[source]

Returns a variable representing the comparison of `a` elements with nan.

This is equivalent to `numpy.isnan`.

`aesara.tensor.``isinf`(a)[source]

Returns a variable representing the comparison of `a` elements with inf or -inf.

This is equivalent to `numpy.isinf`.

`aesara.tensor.``isclose`(a, b, rtol=1e-05, atol=1e-08, equal_nan=False)[source]

Returns a symbolic `'int8'` tensor representing where two tensors are equal within a tolerance.

The tolerance values are positive, typically very small numbers. The relative difference `(rtol * abs(b))` and the absolute difference `atol` are added together to compare against the absolute difference between `a` and `b`.

For finite values, isclose uses the following equation to test whether two floating point values are equivalent: `|a - b| <= (atol + rtol * |b|)`

For infinite values, isclose checks if both values are the same signed inf value.

If equal_nan is True, isclose considers NaN values in the same position to be close. Otherwise, NaN values are not considered close.

This is equivalent to `numpy.isclose`.

`aesara.tensor.``allclose`(a, b, rtol=1e-05, atol=1e-08, equal_nan=False)[source]

Returns a symbolic `'int8'` value representing if all elements in two tensors are equal within a tolerance.

See notes in `isclose` for determining values equal within a tolerance.

This is equivalent to `numpy.allclose`.

### Condition¶

`aesara.tensor.``switch`(cond, ift, iff)[source]

Returns a variable representing a switch between ift (i.e. “if true”) and iff (i.e. “if false”) based on the condition cond. This is the Aesara equivalent of `numpy.where`.

Parameter: cond - symbolic Tensor (or compatible) ift - symbolic Tensor (or compatible) iff - symbolic Tensor (or compatible) symbolic Tensor
```import aesara.tensor as at
a,b = at.dmatrices('a','b')
x,y = at.dmatrices('x','y')
z = at.switch(at.lt(a,b), x, y)
```
`aesara.tensor.``where`(cond, ift, iff)[source]

Alias for `switch`. where is the NumPy name.

`aesara.tensor.``clip`(x, min, max)[source]

Return a variable representing `x`, but with all elements greater than `max` clipped to `max` and all elements less than `min` clipped to `min`.

Normal broadcasting rules apply to each of `x`, `min`, and `max`.

Note that there is no warning for inputs that are the wrong way round (`min > max`), and that results in this case may differ from `numpy.clip`.

### Bit-wise¶

The bitwise operators possess this interface:
Parameter: a - symbolic tensor of integer type. b - symbolic tensor of integer type.

Note

The bitwise operators must have an integer type as input.

The bit-wise not (invert) takes only one parameter.

Return type: symbolic tensor with corresponding dtype.
`aesara.tensor.``and_`(a, b)[source]

Returns a variable representing the result of the bitwise and.

`aesara.tensor.``or_`(a, b)[source]

Returns a variable representing the result of the bitwise or.

`aesara.tensor.``xor`(a, b)[source]

Returns a variable representing the result of the bitwise xor.

`aesara.tensor.``invert`(a)[source]

Returns a variable representing the result of the bitwise not.

`aesara.tensor.``bitwise_and`(a, b)[source]

Alias for `and_`. bitwise_and is the NumPy name.

`aesara.tensor.``bitwise_or`(a, b)[source]

Alias for `or_`. bitwise_or is the NumPy name.

`aesara.tensor.``bitwise_xor`(a, b)[source]

Alias for `xor_`. bitwise_xor is the NumPy name.

`aesara.tensor.``bitwise_not`(a, b)[source]

Alias for invert. invert is the NumPy name.

Here is an example using the bit-wise `and_` via the `&` operator:

```import aesara.tensor as at
x,y = at.imatrices('x','y')
z = x & y
```

### Mathematical¶

`aesara.tensor.``abs`(a)[source]

Returns a variable representing the absolute of `a`, i.e. `|a|`.

Note

Can also be accessed using `builtins.abs`: i.e. `abs(a)`.

`aesara.tensor.``angle`(a)[source]

Returns a variable representing angular component of complex-valued Tensor `a`.

`aesara.tensor.``exp`(a)[source]

Returns a variable representing the exponential of `a`.

`aesara.tensor.``maximum`(a, b)[source]

Returns a variable representing the maximum element by element of a and b

`aesara.tensor.``minimum`(a, b)[source]

Returns a variable representing the minimum element by element of a and b

`aesara.tensor.``neg`(a)[source]

Returns a variable representing the negation of `a` (also `-a`).

`aesara.tensor.``reciprocal`(a)[source]

Returns a variable representing the inverse of a, ie 1.0/a. Also called reciprocal.

`aesara.tensor.``log`(a), log2(a), log10(a)[source]

Returns a variable representing the base e, 2 or 10 logarithm of a.

`aesara.tensor.``sgn`(a)[source]

Returns a variable representing the sign of a.

`aesara.tensor.``ceil`(a)[source]

Returns a variable representing the ceiling of a (for example ceil(2.1) is 3).

`aesara.tensor.``floor`(a)[source]

Returns a variable representing the floor of a (for example floor(2.9) is 2).

`aesara.tensor.``round`(a, mode='half_away_from_zero')[source]
Returns a variable representing the rounding of a in the same dtype as a. Implemented rounding mode are half_away_from_zero and half_to_even.
`aesara.tensor.``iround`(a, mode='half_away_from_zero')[source]

Short hand for cast(round(a, mode),’int64’).

`aesara.tensor.``sqr`(a)[source]

Returns a variable representing the square of a, ie a^2.

`aesara.tensor.``sqrt`(a)[source]

Returns a variable representing the of a, ie a^0.5.

`aesara.tensor.``cos`(a), sin(a), tan(a)[source]

Returns a variable representing the trigonometric functions of a (cosine, sine and tangent).

`aesara.tensor.``cosh`(a), sinh(a), tanh(a)[source]

Returns a variable representing the hyperbolic trigonometric functions of a (hyperbolic cosine, sine and tangent).

`aesara.tensor.``erf`(a), erfc(a)[source]

Returns a variable representing the error function or the complementary error function. wikipedia

`aesara.tensor.``erfinv`(a), erfcinv(a)[source]

Returns a variable representing the inverse error function or the inverse complementary error function. wikipedia

`aesara.tensor.``gamma`(a)[source]

Returns a variable representing the gamma function.

`aesara.tensor.``gammaln`(a)[source]

Returns a variable representing the logarithm of the gamma function.

`aesara.tensor.``psi`(a)[source]

Returns a variable representing the derivative of the logarithm of the gamma function (also called the digamma function).

`aesara.tensor.``chi2sf`(a, df)[source]

Returns a variable representing the survival function (1-cdf — sometimes more accurate).

C code is provided in the Theano_lgpl repository. This makes it faster.

https://github.com/Theano/Theano_lgpl.git

## Linear Algebra¶

`aesara.tensor.``dot`(X, Y)[source]
For 2-D arrays it is equivalent to matrix multiplication, and for 1-D arrays to inner product of vectors (without complex conjugation). For N dimensions it is a sum product over the last axis of a and the second-to-last of b:
Parameters: X (symbolic tensor) – left term Y (symbolic tensor) – right term symbolic matrix or vector the inner product of `X` and `Y`.
`aesara.tensor.``outer`(X, Y)[source]
Parameters: X (symbolic vector) – left term Y (symbolic vector) – right term symbolic matrix vector-vector outer product
`aesara.tensor.``tensordot`(a, b, axes=2)[source]

Given two tensors a and b,tensordot computes a generalized dot product over the provided axes. Aesara’s implementation reduces all expressions to matrix or vector dot products and is based on code from Tijmen Tieleman’s `gnumpy` (http://www.cs.toronto.edu/~tijmen/gnumpy.html).

Parameters: a (symbolic tensor) – the first tensor variable b (symbolic tensor) – the second tensor variable axes (int or array-like of length 2) – an integer or array. If an integer, the number of axes to sum over. If an array, it must have two array elements containing the axes to sum over in each tensor. Note that the default value of 2 is not guaranteed to work for all values of a and b, and an error will be raised if that is the case. The reason for keeping the default is to maintain the same signature as NumPy’s tensordot function (and np.tensordot raises analogous errors for non-compatible inputs). If an integer i, it is converted to an array containing the last i dimensions of the first tensor and the first i dimensions of the second tensor: axes = [range(a.ndim - i, b.ndim), range(i)] If an array, its two elements must contain compatible axes of the two tensors. For example, [[1, 2], [2, 0]] means sum over the 2nd and 3rd axes of a and the 3rd and 1st axes of b. (Remember axes are zero-indexed!) The 2nd axis of a and the 3rd axis of b must have the same shape; the same is true for the 3rd axis of a and the 1st axis of b. a tensor with shape equal to the concatenation of a’s shape (less any dimensions that were summed over) and b’s shape (less any dimensions that were summed over). symbolic tensor

It may be helpful to consider an example to see what tensordot does. Aesara’s implementation is identical to NumPy’s. Here a has shape (2, 3, 4) and b has shape (5, 6, 4, 3). The axes to sum over are [[1, 2], [3, 2]] – note that a.shape == b.shape and a.shape == b.shape; these axes are compatible. The resulting tensor will have shape (2, 5, 6) – the dimensions that are not being summed:

```import numpy as np

a = np.random.random((2,3,4))
b = np.random.random((5,6,4,3))

c = np.tensordot(a, b, [[1,2],[3,2]])

a0, a1, a2 = a.shape
b0, b1, _, _ = b.shape
cloop = np.zeros((a0,b0,b1))

# Loop over non-summed indices--these exist in the tensor product
for i in range(a0):
for j in range(b0):
for k in range(b1):
# Loop over summed indices--these don't exist in the tensor product
for l in range(a1):
for m in range(a2):
cloop[i,j,k] += a[i,l,m] * b[j,k,m,l]

assert np.allclose(c, cloop)
```

This specific implementation avoids a loop by transposing a and b such that the summed axes of a are last and the summed axes of b are first. The resulting arrays are reshaped to 2 dimensions (or left as vectors, if appropriate) and a matrix or vector dot product is taken. The result is reshaped back to the required output dimensions.

In an extreme case, no axes may be specified. The resulting tensor will have shape equal to the concatenation of the shapes of a and b:

```>>> c = np.tensordot(a, b, 0)
>>> a.shape
(2, 3, 4)
>>> b.shape
(5, 6, 4, 3)
>>> print(c.shape)
(2, 3, 4, 5, 6, 4, 3)
```
Note: See the documentation of numpy.tensordot for more examples.
`aesara.tensor.``batched_dot`(X, Y)[source]
Parameters: x – A Tensor with sizes e.g.: for 3D (dim1, dim3, dim2) y – A Tensor with sizes e.g.: for 3D (dim1, dim2, dim4)

This function computes the dot product between the two tensors, by iterating over the first dimension using scan. Returns a tensor of size e.g. if it is 3D: (dim1, dim3, dim4) Example:

```>>> first = at.tensor3('first')
>>> second = at.tensor3('second')
>>> result = batched_dot(first, second)
```
Note: This is a subset of `numpy.einsum`, but we do not provide it for now. X (symbolic tensor) – left term Y (symbolic tensor) – right term tensor of products
`aesara.tensor.``batched_tensordot`(X, Y, axes=2)[source]
Parameters: x – A Tensor with sizes e.g.: for 3D (dim1, dim3, dim2) y – A Tensor with sizes e.g.: for 3D (dim1, dim2, dim4) axes (int or array-like of length 2) – an integer or array. If an integer, the number of axes to sum over. If an array, it must have two array elements containing the axes to sum over in each tensor. If an integer i, it is converted to an array containing the last i dimensions of the first tensor and the first i dimensions of the second tensor (excluding the first (batch) dimension): ```axes = [range(a.ndim - i, b.ndim), range(1,i+1)] ``` If an array, its two elements must contain compatible axes of the two tensors. For example, [[1, 2], [2, 4]] means sum over the 2nd and 3rd axes of a and the 3rd and 5th axes of b. (Remember axes are zero-indexed!) The 2nd axis of a and the 3rd axis of b must have the same shape; the same is true for the 3rd axis of a and the 5th axis of b. a tensor with shape equal to the concatenation of a’s shape (less any dimensions that were summed over) and b’s shape (less first dimension and any dimensions that were summed over). tensor of tensordots

A hybrid of batch_dot and tensordot, this function computes the tensordot product between the two tensors, by iterating over the first dimension using scan to perform a sequence of tensordots.

Note: See `tensordot()` and `batched_dot()` for supplementary documentation.
`aesara.tensor.``mgrid`()[source]
Returns: an instance which returns a dense (or fleshed out) mesh-grid when indexed, so that each returned argument has the same shape. The dimensions and number of the output arrays are equal to the number of indexing dimensions. If the step length is not a complex number, then the stop is not inclusive.

Example:

```>>> a = at.mgrid[0:5, 0:3]
>>> a.eval()
array([[0, 0, 0],
[1, 1, 1],
[2, 2, 2],
[3, 3, 3],
[4, 4, 4]])
>>> a.eval()
array([[0, 1, 2],
[0, 1, 2],
[0, 1, 2],
[0, 1, 2],
[0, 1, 2]])
```
`aesara.tensor.``ogrid`()[source]
Returns: an instance which returns an open (i.e. not fleshed out) mesh-grid when indexed, so that only one dimension of each returned array is greater than 1. The dimension and number of the output arrays are equal to the number of indexing dimensions. If the step length is not a complex number, then the stop is not inclusive.

Example:

```>>> b = at.ogrid[0:5, 0:3]
>>> b.eval()
array([,
,
,
,
])
>>> b.eval()
array([[0, 1, 2]])
```

`aesara.gradient.``grad`(cost, wrt, consider_constant=None, disconnected_inputs='raise', add_names=True, known_grads=None, return_disconnected='zero', null_gradients='raise')[source]
For more information about how automatic differentiation works in Aesara, see `gradient`. For information on how to implement the gradient of a certain Op, see `grad()`.
Parameters: cost (`Variable` scalar (0-dimensional) tensor variable or `None`) – Value that we are differentiating (that we want the gradient of). May be `None` if `known_grads` is provided. wrt (`Variable` or list of Variables) – Term[s] with respect to which we want gradients consider_constant (list of variables) – Expressions not to backpropagate through disconnected_inputs ({'ignore', 'warn', 'raise'}) – Defines the behaviour if some of the variables in `wrt` are not part of the computational graph computing `cost` (or if all links are non-differentiable). The possible values are: ’ignore’: considers that the gradient on these parameters is zero. ’warn’: consider the gradient zero, and print a warning. ’raise’: raise DisconnectedInputError. add_names (bool) – If True, variables generated by grad will be named (d/d) provided that both cost and wrt have names known_grads (OrderedDict, optional) – A ordered dictionary mapping variables to their gradients. This is useful in the case where you know the gradient on some variables but do not know the original cost. return_disconnected ({'zero', 'None', 'Disconnected'}) – ‘zero’ : If wrt[i] is disconnected, return value i will be wrt[i].zeros_like() ’None’ : If wrt[i] is disconnected, return value i will be None ’Disconnected’ : returns variables of type DisconnectedType null_gradients ({'raise', 'return'}) – Defines the behaviour if some of the variables in `wrt` have a null gradient. The possibles values are: ’raise’ : raise a NullTypeGradError exception ’return’ : return the null gradients Symbolic expression of gradient of `cost` with respect to each of the `wrt` terms. If an element of `wrt` is not differentiable with respect to the output, then a zero variable is returned. variable or list/tuple of variables (matches `wrt`)